Undefined for Different Reasons

harpazo65 · 2024-09-09 02:34:03

Nothing a/0 (where a does not equal 0) and 0/0 are undefined, but for different reasons. Explain the different reasons.

Oculus8596 · 2024-09-18 16:29:15

What have you done so far?

Bob · 2024-09-19 00:19:02

Division is defined as the inverse process to multiplication.

So, for example, we know that 6 x 8 = 48. This leads to 48 ÷ 8 = 6.

To attach a meaning to 48 divided by 8 we can ask "What number times 8 gives 48?"

So, a/0 would mean asking "What number times 0 gives a?" You cannot find a number to answer this question.

Instead of 0, you could consider a very small number instead of 0. Let's choose a = 6 and divide by 0.1. We get an answer of 60.

Divide by 0.01 and we get 600. Divide by 0.001 and we get 6000. So as the divisor gets smaller the answer gets bigger so we might say as the divisor tends to zero the answer tends to infinity. This is useful when trying to sketch a graph.

What about 0 ÷ 0 ?

For this the question would be "What number times 0 gives 0?" This time there's no shortage of answers as every number times 0 gives 0. So we have to say the answer is indeterminate.

In differential calculus for a general point (x,y) we construct a chord by joining {x,f(x)} to {x + h, f(x+h)} and calculate it's gradient: {f(x+h) - f(x)} / {h}

As h tends to zero this gradient may tend to a limit and that enables us to assume the gradient at the point is that limit. It works for lots of functions, so in those special circumstances 0/0 can be given a value.

Bob

Oculus8596 · 2024-09-20 12:12:54

Bob wrote:

Division is defined as the inverse process to multiplication.
So, for example, we know that 6 x 8 = 48. This leads to 48 ÷ 8 = 6.
To attach a meaning to 48 divided by 8 we can ask "What number times 8 gives 48?"
So, a/0 would mean asking "What number times 0 gives a?" You cannot find a number to answer this question.
Instead of 0, you could consider a very small number instead of 0. Let's choose a = 6 and divide by 0.1. We get an answer of 60.
Divide by 0.01 and we get 600. Divide by 0.001 and we get 6000. So as the divisor gets smaller the answer gets bigger so we might say as the divisor tends to zero the answer tends to infinity. This is useful when trying to sketch a graph.
What about 0 ÷ 0 ?
For this the question would be "What number times 0 gives 0?" This time there's no shortage of answers as every number times 0 gives 0. So we have to say the answer is indeterminate.
In differential calculus for a general point (x,y) we construct a chord by joining {x,f(x)} to {x + h, f(x+h)} and calculate it's gradient: {f(x+h) - f(x)} / {h}
As h tends to zero this gradient may tend to a limit and that enables us to assume the gradient at the point is that limit. It works for lots of functions, so in those special circumstances 0/0 can be given a value.
Bob

Here's my take on it:

1. a/0 (where a ≠ 0): Division as Repeated Subtraction.
The Concept: Division can be thought of as repeated subtraction. For example, 12 / 4 asks “How many times can we subtract 4 from 12 until we reach 0?”. The answer is 3. The Problem: If you try this with a/0, you run into a wall. How many times can you subtract 0 from ‘a’ to reach 0?

You can subtract 0 infinitely, and you’ll never actually reach 0. This leads to an infinitely large result, which is undefined in standard arithmetic. Do you agree? If not, why not?

2. 0/0: The Ambiguity Problem
The Concept: Let’s think about division in terms of multiplication. If you have the equation a / b = c, it can be rewritten as a = b * c.

The Problem: With 0/0, we’re essentially asking: “What number, when multiplied by 0, equals 0?” Here’s the catch – any number multiplied by 0 equals 0.

This means, as I see it, there’s no single, unique answer. The result is ambiguous, and therefore, undefined.

My Conclusion:

a/0 (a ≠ 0) is undefined because it represents an infinitely large value.

0/0 is undefined because it represents an ambiguous value, with infinitely many possible solutions.

Do you agree?

Bob · 2024-09-20 19:24:21

1. I'm happy with your repeated subtraction argument. As multiplication can be thought of as repeated addition my method and yours have the same root. Yours gets to infinity more neatly so I'll adpot it in the future.

2. This is exactly my argument.

Bob

Oculus8596 · 2024-09-20 21:11:18

Bob wrote:

1. I'm happy with your repeated subtraction argument. As multiplication can be thought of as repeated addition my method and yours have the same root. Yours gets to infinity more neatly so I'll adpot it in the future.
2. This is exactly my argument.
Bob

We both reached the same correct street, so to speak.

Math Is Fun Forum

#1 2024-09-09 02:34:03

Undefined for Different Reasons

#2 2024-09-18 16:29:15

Re: Undefined for Different Reasons

#3 2024-09-19 00:19:02

Re: Undefined for Different Reasons

#4 2024-09-20 12:12:54

Re: Undefined for Different Reasons

#5 2024-09-20 19:24:21

Re: Undefined for Different Reasons

#6 2024-09-20 21:11:18

Re: Undefined for Different Reasons

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